L^p norm - Knowlpedia

A $L^p$ norm on a measure space $(X,\Sigma,\mu)$ (for $1\le p<\infty$ ) assigns to a measurable function $f:X\to\mathbb{R}$ the value

\|f\|_p := \left(\int_X |f(x)|^p\,d\mu(x)\right)^{1/p},

whenever the Lebesgue integral of the nonnegative function $|f|^p$ is finite; here $|f(x)|$ uses the absolute value. For $p=\infty$ one defines

\|f\|_\infty := \operatorname*{ess\,sup}_{x\in X} |f(x)|,

using the essential supremum.

If $f$ and $g$ are equal almost everywhere, then $\|f\|_p=\|g\|_p$ , so the $L^p$ norm is naturally a norm on the corresponding $L^p$ space.

Examples

On $([0,1],\mathcal{B},\lambda)$ , the constant function $f(x)=1$ satisfies $\|f\|_p=1$ for every $1\le p\le\infty$ .
On $([0,1],\mathcal{B},\lambda)$ , for $f(x)=x$ one has $\|f\|_p=(1/(p+1))^{1/p}$ for $1\le p<\infty$ , and $\|f\|_\infty=1$ .

Complex and vector-valued functions

The same formulas apply to complex functions using the complex modulus, and to finite-dimensional vector-valued functions using their Euclidean norm. Allowing the displayed integral to be infinite defines an extended size on all measurable functions; the finite-size classes are the Lebesgue spaces.